Microwave polarization apparatus



Dec. 30, 1958 T. N. ANDERSON 2,865,972

MICROWAVE POLARIZATION APPARATUS Filed Sept. 26, 1956 s Sheets-Sheet 1 IN V EN TOR. Zaire JV. Avaeraozy,

Dec. 30, 1958 Filed Sept. 26, 1956 PERMEABILITY T. N. ANDERSON MICROWAVE POLARIZATIQN APPARATUS 5 Sheets-Sheet 2 moo 2000 .3000 4000 MAGNETIC FLUX FIELD (a uss) INVENTOR. 752? /V. A 7762 61150 7 wmwa /ym;

Aida ragga Dec. 30, 1958 T. N. ANDERSON MICROWAVE POLARIZATION APPARATUS 3 Sheets-Sheet 5 Filed Sept. 26, 1956 INVENTOR.

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fllzirve Unite 2,866,972 MICROWAVE PoLAnrzArro APPARATUS Tore N. Anderson, Mountainside, N. 1., assignor to Airtron, Inc., Linden,'N. J;, a corporation of New Jersey This invention relates to microwave waveguides and particularly to a magneticallyscontrolled waveguide trans former. The microwave transformer of my invention is adapted to convert plane-polarized microwaves't'o waves ofcircular, elliptical or linear polarized field 'modes of either 905" displaced orientation. The circular and elliptical waves may be polarized in either a right or' leftlhand direction. i

In the microwave transformer constructed in accordance with my invention, no moving parts are necessary to" control the polarization of the microwave output energy. The polarization is controlled'magnetically by the use of ferrite slabs positioned in the waveguide transformer. With plane-polarized energy input to the transformer, the polarization of the outputfrom the transformer is dependent upon the strength of an external, unidirectional, magnetic flux field which is .appliedt'o' the ferrite slabs. For example, a plane-polarized wave' having equal space quadrature components, which are in phase in time may be converted to a wave of either 90 displaced orientation, or to a circularly or elliptically polarized wave of either sense; the polarization of the output wave being dependent only upon the strength of the magnetic field applied'to the ferrite slabs. It is seen, then, that the microwave transformer of my invention is of a type which may. be used in many difierent and varied applications, and which is easily and simply controlled magnetically.

Any electrical linearly polarized wave may be resolved into two equal components which are 90.a'part in space but together in time. A circularly polarized wave results, in the special case, when the two equal space quadrature components are also spaced 90 apart electrically (that is, in time). The sense of .the circularly polarized wave depends upon whether the one 'space quadrature component lags the other component .903 ,or leads by 90.

'If the two space quadrature vectors are .not ofequal amplitude, or if the time phase difference between .them is'other than 90. the wave is ellipticallyypolarized; the sense again depending whether one component leadslor lags the other. 'In the special case where the space quadrature components are 180 out of phase, a planepolarized .wave results. The resulting plane-polarized wave is orthogonally positioned with respect to the planepolarized wave which results when the space quadrature components are in .time phase. 7

It is seen, then, that any right or-left hand circularly or elliptically polarized wave, as well as linearly polarized waves of either orthogonal orientation, may be obtained by acting upon one of two equal quadrature components so as to change the time relation therebetween. With the apparatus of my invention, variable amounts of time phase-shift may be introduced between two microwave components 90 apart in space; the amount of the time phase-shift being magnetically controlled.

An'object of this invention is the provisionvof an apparatus for producing circularly or elliptically polarized microwaves of either sense,'or linearly polarizedrnicrowaves of either orthogonal orientation, from plane-polarized'microwaves.

States Patent" C 7-27 of Figure 5,

f 2,865,972 Pain D -3 1?? An object of this invention is the provision of a device for transforming two equal, spaced quadrature microwave components to any desired circular, elliptical or linear mode polarized wave. An object of this invention is the provision of a magnetically controlled device for converting equal and spaced quadrature microwave components to circularly, elliptically or linearly polarized waves of any type upon variation of a magnetic flux field applied thereto.

An object of this invention is the provision of a waveguide device comprising a section of waveguide of a generally cruciform-shaped cross-section and which is provided with phase shifting elements comprising slabspf ferrite material positioned in two diagonally oppositely spaced arms thereof, and an unidirectional magnetic flux field, the strength of which maybe varied. An object of this invention is the provision of a device for introducing in microwave energy selectively"variable amounts of time phase displacement between thespace quadrature components of the plane-polarized waves 'of the said energy, the said device comprising a hollow waveguide section of generally cruciform-shaped cross-section, a slab of ferrite material fastened to the interior of each side wall of two diagonally opposed arms of the said cruciform-shaped waveguide, means impressing a transverse unidirectional magnetic flux field through the said slabs of'ferrite, and means varying the strength of the said magnetic flux field.

An object of this invention is the provision of a beam switching antenna feed system in which plane-polarized waves may be switched to circularly or elliptically polarized waves of either orthogonal orientation.

These and other objects and advantages Will become apparent from the following description when taken with the accompanying drawings. It will be understood the drawings are for purposes of illustration and are not to be construed as defining the scope or limits of the invention, reference being had for the latter purpose to'the appended claim.

Figure l is an end view of a microwave transformer made in accordance with my invention,

Figure 2 is a vertical longitudinal sectional view taken along line 22 of Figure 1,

Figure 3 is a horizontal longitudinal section view taken along line 33 of Figure 1,

Figure 4 is a typical permeability versus magnetic flux field curve for ferrite material used as phase shifting elements in my device,

Figure 5 is an antenna feed system embodying my novel transformer device,

Figure 6 is a transverse sectional view taken along line 6-6 of Figure 5,

Figure 7 is a transverse sectional view taken alongline Figure 8 is a transverse sectional view taken along line 88 of Figure 5,

Figure9 is a transverse sectional view taken along line 99 of Figure 5, and c Figures 10-15 illustrate the electric field patterns presentat lines l010 through 15-l5, respectively, along the antenna feed system of Figure 5. J

Referring to Figure 1 of the drawings, my novel transformer device is shown comprising a hollow waveguide 20 of a cruciform-shaped cross-section, phase shifter elements 21, electromagnets 22 and 22' anda source'of variable voltage 23. The cruciform-shaped waveguide 20 comprises U-shaped horizontally extending arms which includes side walls 25 and 25 and end walls 26'and: 26', respectively. Similar vertically extending U-shaped arms include side walls27 and 27' and end walls'28 and "28, respectively. .The cruciform-shaped waveguide 20K is adapted to transmit plane-polarized microwave energy: in

3 space quadrature therethrough. The horizontally extending U-shaped arms support vertically polarized energy as indicated by the voltage vectors E while the vertically substantially equal horizontally and vertically polarized components. Any method may be used to produce such a field having substantially equal amplitude components 90 apart in space in the cruciform-shaped waveguide 20. For example, the two components may be introduced separately into the waveguide or, they may be generated by applying a linearly polarized wave from a rectangularshaped waveguide diagonally across a square-shaped waveguide. The resulting electric vector will have substantially equal components perpendicular to each pair of sides of the square-shaped waveguide, and these perpendicularly-spaced components may be applied to the cruciform-shaped waveguide 20 at one end indicated by the arrow 29.

Various magnetic materials may be used for the phase shifter elements 21. Ferrites, and particularly ferrites having a cubic crystal structure, are very effective. Ferrites comprising zinc oxide, ferric oxide and manganese oxide which has been sintered, ground and pressed or molded into the form of the phase shifting elements 21 give good results. The phase shifting elements 21 are in the form of comparatively thin, flat, longitudinally positioned slabs which are mounted securely against the side walls of two diagonally opposite arms of the cruciformshaped waveguide. In the drawings (Figures 1-3) the phase shifting elements 21 are shown mounted in the horizontally extending U-shape of the waveguide 20. Due to the symmetry of the waveguide 20, it will be apparent that the phase shifter elements could be mounted in the vertically extending U-shaped arms instead, and the electromagnets positioned on the vertical extending U-shaped arms.

The phase shifting elements 21 may be mounted in diagonally opposite arms in any suitable manner and position. Only one possible mounting position is shown in the drawings; the illustrated position being preferred, particularly when large amounts of power are to be transmitted through the waveguide 20. The wide side of the phase shifting elements 21 are mounted flat against the side walls 25, 25' while the narrow side of each element 21.abuts, and is joined to, the end walls 26, 26'. With this mounting position it will be apparent that the heat generated in the elements 21 is easily conducted to the walls of the waveguide 20, from which walls it may be radiated into the surrounding atmosphere.

It will be understood that other mounting positions for the ferrite phase shifting elements 21 may be used. By way of example, only two phase shifting elements of the same general shape as those shown in the drawings may be used. One element could then be mounted longitudinally in opposite diagonally spaced arms of the waveguide 20 with the wide side parallel to the end walls 26, 26 and spaced therefrom. It will be apparent that the amount of heat which may be dissipated by the elements with such mounting is less than that for the elements mounted in the preferred manner as shown in Figures 1-3. It will be understood also that the phase shifting elements 21 as shown in Figures 1-3 need not extend to the end walls 26 and 26' but may be located a spaced distance therefrom. Further, although the ends of the phase shifting elements 21 are shown to be fiat in the drawings, it will be understood that they may be tapered in order to reduce reflection of energy transmitted through the waveguide 20.

A magnetic field, parallel to the plane of polarization of the microwave energy indicated by the voltage vectors E is established through the phase shifter elements 21 by use of the electromagnets 22 and 22'. The electro- '4 magnets comprise pole pieces 31, 32 and 31, 32' which are joined by yokes 33 and 33, respectively; all made of low reluctance magnetic material. Each yoke member 33 and 33 is provided with a coil winding 34 and 34', respectively. The pole pieces 31, 32 and 31', 32' are placed close to the side walls 25, 25 0f the waveguide 20 to reduce leakage flux to a minimum. The windings 34, 34 are connected in series circuit connection with a source of variable voltage 23 which comprises a potentiometer 36 and a parallel connected battery 37, or other source of constant potential. One end of the parallel connected potentiometer 36 and battery 37 is connected through a lead wire 39 to the coil winding 34,

while the adjustable arm 41 of the potentiometer 36 is connected through the lead wire 42 to the other coil winding 34'. The two coil windings 34 and 34 are connected together by a lead wire 43 to complete the coil winding energizing circuit.

i It is seen, then, that the electromagnets 22 and 22', together with the variable voltage source 23, provide an unidirectional magnetic flux field, of adjustable strength, through the phase shifting elements 21; the direction of the magnetic flux field being parallel to the voltage vectors E in the two horizontally extending U-shaped arms of the cruciform-shaped waveguide 20. It will be understood that a unidirectional magnetic flux field through the phase shifting elements 21 may be created by other means as, for example, permanent magnets which are located in the same positions as the electromagnets 22 and 22'. The amount of the field flux through the phase shifting elements 21 could be controlled by varying the position of the permanent magnets with respect to the phase shifting elements. Obviously, a combination of permanent magnets and electromagnets could be used, if desired.

The output end of the cruciform-shaped waveguide section 20 (that is, the end opposite the arrow 29) may be terminated in any suitable load, or connected to waveguide sections having a cross-sectional shape other than cruciform by the use of appropriate waveguide transition sections. For example, a waveguide of square-shaped cross-section may be suitably secured to the cruciformshaped waveguide 20 and thereby provide means whereby linear polarized microwave energy in either displaced orientation may be supported. If my novel transformer device is to be used to transform a linearly polarized wave to either a circularly or elliptically polarized wave, a waveguide which supports these modes, as for example, a circular-shaped waveguide, could be used at the output end of the waveguide section 20. A circular-shaped waveguide section attached to the output end of my novel transformer device will also support linearly-polarized waves and is often used in applications where a linear, circular or elliptical mode is to be employed at various times.

A typical permeability versus unidirectional magnetic flux field curve for a ferrite phase shifting element, such as any of the phase shifting elements 21, is shown in Figure 4 of the drawings. Any permeability versus magnetic flux field curve for a ferrite element is dependent upon the frequency of the microwave energy to which the ferrite element is subjected, and the exact composition of the ferrite element. The curve shown in Figure 4 is typical of a ferrite member comprising zinc oxide, ferric oxide and manganese oxide and which is subjected to microwave energy having a wavelength of three centimeters. Referring to the curve of Figure 4, it is seen that upon application of a unidirectional magnetic fim: field of from 0 to 1000 gauss, the permeability is positive; beginning at approximately unity at zero magnetic flux field and decreasing to zero permeability at approximate- 1y 1000 gauss. Between 1000 and 3000 gauss, approximately, the permeability of the ferrite material is negative, and above 3000 gauss returns'to apositive permeability. Ferromagnfiilc resonance occurs at a-magnetic iflu r field'of approximately 3000 gau'ss and accounts for theincreased values of permeability at that point.

i As mentioned above in the description of Figures 1-3, the inp ut to the cruciform-shapedwaveguide 20 comprises two substantially equal space quadrature components which are in time phase relation; the voltage vectors thereof being designated E and E in Figure 1. The ferrite phase shifting elements in the two diagonally opposed and horizontally extending arms of the waveguide 20 act upon the voltage vector E only. The voltage component E in the vertically extending arms is substantially uneffe cted by the ferrite phase shifting elements. if no magnetic flux field is applied to the ferrite phase shifting elements 21, the permeability of the elements is approximately the same as that of the air in the waveguide and the voltage vector E is substantially uneffected. The orthogonal voltage vectors E and E both emerge from the transformer device with the same zero time phase relation with which they were'introduced into the sa d transformer device and may be recombined, by any suitable means, to produce a linearly polarized wave which extends diagonally upwardly and to the right as viewed in Figure l of the drawings.

In order to satisfy the condition for a circularly polarized wave, the two quadrature spaced components E and E must have a time phase of 90 or 270. The wave will be either a right or left handed circularly polarized wave depending Whether the time phase relation is 90 or 270. Either right or left hand circularly-polarized waves may be produced with my transformer device; the sense of the circularly-polarized wave depending upon the strength of the unidirectional magnetic flux field wh ch 18 applied the phase shifting elements 21. By applying a magnetic flux field to the phase shifting elements 21 of somewhat less than 1,000 gauss (within the area marked Region A in Figure 4) wherein the permeability is still positive but less than unity, the phase velocity of the voltage vector E will be decreased. Upon application of the proper magnetic flux field within the Region A, the E voltage vector may be made to lag the E voltage vector by 90 and a right hand circularly polarized output results.

When the magnetic flux field which is applied to the phase shifting elements 21 is further increased, the permeability of the phase shifting elements goes negative as seen in the curve of Figure 4. Now, instead of causing the E voltage vector to lag the E voltage vector, the phase shifting elements cause the E voltage vector to lead. If a proper magnetic flux field is chosen, within the area marked Region B in Figure 4, the E voltage vector will lead by 90 whereupon a left hand circularly polarized wave results at the output of the device. It will be understood that the same results can also be obtained by reversing the applied magnetic field, by reversing the direction of current through the coil windings 34, 34'.

The time phase angle between the two space quadrature components E and E may be increased to 180. When this is done, the resultant .wave is orthogonally positioned with respect to the resultant of the zero time phase quadrature components. A time phase of 180 is accomplished by further increasing the unidirectional magnetic flux field on the phase shifting elements 21 to a point within the area marked Region C in Figure 4. The

ermeability of the phase shifting elements 21 become increasingly negative with an increased magnetic flux field; and by adjusting the magnetic flux field to the proper value within the Region C, the voltage vector E may be made-to lead the voltage vector E by 180.' The resultant linearly polarized wave would extend diagonally downwardly and to the right, as viewed in Figure 1.

Many uses exist for my .novel transformer device. In Figure 5 of the drawings I have shown one application wherein my device is embodied in an antenna feed. system forpurposes of radar beam switching. Aslwill be understood by those skilled in this art, the usefulness of a radar system is increased if the radar beam may be selectively switched to a horizontally,or vertically, or circularly polarized wave. A horizontally'polarized beam is desirable for weather radar purposes since the round return is minimized with horizontal polarization. A V91- tically polarized beam results in greater ground return and would be used for ground mapping purposes and a circularly polarized beam may be used to discriminate against rain drops.

Referring now to Figure 5, only the waveguide portion of the antenna system is shown; no source of microwave energy or beam concentrating reflector is shown. Microwave energy (from the source not shown) is fed into a rectangular-shaped waveguide section 50 in the direction of the arrow 51 in a TB mode, preferably. The rectangular-shaped waveguide section 50 is coupled, by-the usual flanges 52 and 53 and fastening means '54, to a transformer and power splitter 56 wherein the microwave energy from the rectangular-shaped waveguideSl) is split into two equal and orthogonally positioned components. As seen in Figure 6, the rectangular to square waveguide transformer and power splitter is provided with a generally diamond-shaped aperture 57 at the flange 53 and terminates as seen in Figure 7, in asquare-shaped aperture 59 in the flange 58 at the other end thereof.

The rectangular to square waveguide transformer and power splitter 56 is coupled, by the usual flanges 58 and 59 and fastening means 60, .to a second transformer 62 wherein the equal and orthogonally positioned microwave energy components are fed to a cruciform-shaped waveguide section 63. A cross-sectional view of the square to cruciform-shaped transformer is shown in Figure 8 of the drawings and, as seen in'Figure 8, the transition from square to cruciform shape is accomplished with the use of tapered walls 64 within the transformer 62, as will be understood by those skilled in this art. The cruciformshaped waveguide section 63 is provided with phase shifting elements (not visible in Figure 5) and electromagnets indicated generally by reference numeral 64; which were described above in detail and shown in-Figures 1-3. The lead wires 39 and 42' are connected to the electromagnet coils in the transformer device and are adapted to be'connected to a variable source of D.-C. current, not'shown in Figure 5.

The output end of my novel transformer device is connected to a circular-shaped waveguide section 66 through the cruciform to circular transition 67. A'crosssectional View of the transformer 67 is shown in Figure 9. of the drawings. A linear taper transition is made from the circular-shaped waveguide section 66 to a solid dielectric 68 as is best seenin the portion which is broken away in Figure 5. A linear taper, as at'69, is made .in order ,to reduce the impedance mismatch between the air and solid dielectric. The solid dielectric terminates .in a circularly symmetrical lens 71 which forms a part of a ring focus feed 72. Included in the ring focus feed 69 is a primary reflector 73 which is suitably attached to the .dielectric lens 71. The antenna feed system "shown in Figure 5 projects through thevertex of a beam concentrating reflector (not shown in the drawings) such that the dielectric ring focus feed 72 is positioned at the focal point of the beam concentrating reflector. The dielectric ring focus feed 72 provides a substantially pointsource of energy which is directed toward the beam shaping refiector as is well understood; the resulting beam shape depending upon the shape. of the beam focusing reflector. A beam shaping reflector in the .form of a paraboloid is often used. I

Reference is now made to Figures 10 through 15 of the drawings wherein the electric field pattern at various points along the antenna .feed'systm are'shown. The

diagrams are not intended to represent the actual crosssectional ,views' of the antenna system, but rather illustrate schematically the'internal shape of the 'waveguide 'and the electric field vectors therein. The microwave energy is introduced into the rectangular-shaped waveguide section 50 in a TE mode, preferably, and is vertically polarized as shown in Figure 10. In the rectangular to square waveguide transformer and power splitter 56, the vertically polarized wave, shown in Figure 11, is transformed into two equal voltage components E and E which are orthogonally positioned and are in time phase relation, as shown in Figure 12. These orthogonally positioned energy components then pass through the square to cruciform shaped transformer 62 without disruption of the zero time phase relation therebetween, and establish an electric field, as shown in Figure 13, in the cruciformshaped waveguide section 63. My novel transformer device in the cruciform-shaped waveguide section is shown in Figure 14. Phase shifting elements 76 are located in two oppositely disposed arms of the cruciform-shaped waveguide and may be subjected to a unidirectional magnetic flux field provided by the coils 78 and 73. (The pole pieces and yokes have been omitted from the schematic representation of Figure 14 for clarity, as has also the variable source of D.-C. current for energizing the said series connected coils 78 and 78'.)

The different linearly and circularly polarized waves which may be obtained with my novel transformer device are shown in Figure 15 of the drawings. All of the crosssectional views, A through D, of Figure 15 represent various polarization modes obtainable at the circular-shaped waveguide section 66 when different strength magnetic flux fields are applied to the phase shifting elements 76. The energy at the circular-shaped waveguide section 66 is fed to the dielectric ring focus feed 72 and thence to the beam shaping reflector (not shown in the drawings) in the same mode in which the energy appears at the circular-shaped cross-section 66 as seen in Figure 15.

It will be apparent that the energy in the two diagonally opposite arms of the cruciform-shaped waveguide section in which there are no phase shifting elements will pass through the waveguide substantially uneifected by the ferrite phase shifting elements 76 in the other two diagonally opposite arms. That is to say, the phase velocity of the energy represented by the voltage vectors E is uneifected by the phase shifting elements. If the coils 78 and 78' are unenergized, the phase velocity of the energy represented by the voltage vectors E is also substantially uneffected by the phase shifting elements 76 since the permeability of the elements is very close to unity when no unidirection magnetic flux field is applied to them. When no unidirectional magnetic flux field is applied to the phase shifting elements, the two equal and orthogonally positioned voltage vectors time phase relation and are recombined at the circularshaped waveguide 66 so as to produce the vertically polarized vector E as shown in Figure 15A.

Now, if the coils 78 and 78' are energized so that the permeability of the phase shifting elements decreases to some positive value less than unity, the E voltage vector will be caused to lag in time the E voltage vector because of the reduced phase velocity of E If a magnetic flux field of proper strength is produced (within the area designated Region A in Figure 4) a time lag of 90 may be obtained and the condition for a circularly polarized wave results. The resultant circularly polarized wave is shown in Figure 1513.

If additional current is applied to the coils 78 and 78 so that the unidirectional magnetic phase shifting elements is increased the permeability of the elements may be made negative. With a proper magnetic flux field (in the region designated Region B" in Figure 4) the E voltage vector will be caused to lead, in time, the E voltage vector by 90 and again a circularly polarized wave results, as seen in Figure 15C. This time, however, the wave is polarized in the opposite sense as will be apparent upon comparison of Figure 15C with Figure 15B.

The horizontally polarized wave shown in Figure 15D E and E retain their zero flux field through the of the drawings is produced by further increasing the magnetic flux field (within the area designated Region C in Figure 4) to the point that the phase shifting elements 76 cause the voltage vector E to have a time phase relation of with the voltage vector E Again, the condition for a linearly polarized wave is satisfied and the voltage vector E shown in Figure 15D is produced upon combining the voltage vectors E and E (The 180 time phase shift of the E voltage vector is equivalent to a 180 space phase shift and has been shown rotated 188 in space in Figure 15D.) The resultant voltage vector E in Figure 15D is horizontally positioned and so the wave is horizontally polarized.

The diiferent sense 'elliptically polarized waves are produced when the coils 78 and 78' are energized an amount intermediate the amounts necessary to produce the various above described linearly and circularly polarized modes.

My novel antenna system may be used in radar systems wherein one antenna is used for both transmitting and receiving purposes. If circularly or elliptically polarized waves are being transmitted, it becomes necessary when switching back and forth between the transmitting and receiving conditions to also switch the unidirectional magnetic flux field applied to the phase shifting elements. If this is not done, the rectangular-shaped waveguide to the receiver would not support the incoming, or received, wave because the wave would be improperly polarized for transmission therethrough. Switching of the current which is applied to the coils 78 and 78 is easily accomplished electronically.

Having now described my invention in detail in accordance with the patent statutes, various changes and modifications will suggest themselves to those skilled in this art, and it is intended that such changes and modificaiions shall fall within the spirit and scope of the invention as recited in the following claim.

I claim:

An antenna feed system comprising a rectangularshaped waveguide section connected to a source of microwave energy and adapted to support the said energy in a linearly polarized field mode; means splitting the said microwave energy into equal and orthogonally positioned components; a cruciform-shaped waveguide section; means introducing the said equal and orthogonally positioned power components into the said cruciform-shaped waveguide section; means adjusting the time-phase displacement between the said orthogonally positioned energy components in the said cruciformshaped waveguide section, the said means including ferrite slabs in two oppositely disposed arms of the said cruciform-shaped waveguide section and a variable, unidirectional, magnetic fiux field through the said ferrite slabs; a focus feed; waveguide means coupling the said cruciform-shaped waveguide section to the said focus feed, the said focus feed and waveguide means being adapted to support the said microwave energy in linearly and circularly polarized field modes; and a beam shaping reflector adapted to reflect the said microwave energy from the said focus feed.

References Cited in the file of this patent UNITED STATES PATENTS 2,412,320 Carter Dec. 10, 1946 2,441,574 Jaynes May 18, 1948 2,776,412 Sparling Jan. 1, 1957 2,787,765 Fox Apr. 2, 1957 OTHER REFERENCES Bell publication of record, pages 45-46 and 71-86.

Fox et al.: Behavior and Applications of Ferrites in the Microwave Region, The Bell System Technical Journal, vol. 34, No. 1, January 1955, pages 5-103 (page 81 relied on). (Copy in Scientific Library.) 

